Math Intuition Cheatsheet

For a potential math major [i.e. people for whom metric space refers to a measure and not a flat in Europe :)], you definitely want the ground-up understanding. In CS it's similar, where you learn about transistors, logic gates, ALUs, CPUs, machine code, compilers, along with high-level languages.

But, some people just need the HTML "Hello World" to make their webpage. (In the math field, we have students who need calculus primarily to find the min/max of a function, and are wasting time worrying about epsilon-delta definitions of continuity. Limits are interesting, but I'd prefer to ignite curiosity with higher-level topics then dive into the details, instead of forcing someone to learn organic chemistry before being able to drive a car.)

Why would you need to prove central limit theorem in order to use calculus? What kind of course teach it that way? It would sound like something to prove for a honors calculus class that prepares students to become mathematicians.

Anyways, about math education. Math below college are taught by math educators. They are no mathematicians and they do things differently.

Things in college are rigorous. Should college students learn calculus in a way that make engineers happy and mathematicians cringe? "Learning things not from first principles but 'intuition' of the real world? what is this? physics 101? ".

Different professor/department views it differently, depend on the view, you will get different education.

There are places offering business calculus. I personally believe to be much better way to teach calculus to people who don't really need to know how things work, but still need to know how to use it(at least for simple functions appearing in the real world).

> programming as a discipline has learned to avoid; encouraging varied and/or terse notations, opaque variable naming schemes, and arbitrary use of jargon where simpler terms would suffice.

Why should all mathematicians suddenly write things with the exact same convention just so people can learn it easier? Likely, most people who benefits from these won't advance mathematics in anyway.

Here is what I see when you wrote these things, but might sounds more interesting in a programming perspective.

"The barrier of not able to program comes from the number of programming languages, it be nice if we can all agree on the same programming language so I never have to learn another programming language again."

Maybe learning calculus itself is more akin to learning one programming language, but rarely calculus is taught in a way to end all learning(again, depend on professor/department). When mathematicians teach calculus, they would want to teach the principles in order for one to advance even further.

and now to each point(as someone who had pure math education)

1. "encouraging varied notations" This is not true. People might have different preference because which school/field they come from, I have yet to see anyone enjoying people come up with new notations when there are some established notation(s). In general, language use in the same field are mostly consistent (for a few decades, at least). No competent mathematician would have inconsistent notation in one single article.

2. "Terse notations" A few symbols captures a whole lot of ideas. It's not software engineering, the number of different variables/symbols in a paper is not the same order of magnitude as our programs. There are books with table of symbols of around 2 pages, but usually such a book would take years to master.

3. "opaque variable naming schemes", it's a convention can be learned by doing, and you don't even need to follow someone else's scheme. In fact, not knowing the convention doesn't mean anything. Most variables are used a few times and never used again. Just a few days ago I pick up a paper by the Hungarian combinatorial optimization authors, their notation is completely different from what I usually read. Once I read the introduction, I already internalized the "table of variables/notations" so I don't even need to refer back often. To programmers, this is like complaining different open source programs use different naming scheme for their API.

4. "Arbitrary use of jargon where simpler terms would suffice". I do not believe this happens often in mathematics. Mathematicians likes things to be elegance, simple and precise. There must be reasons for us to use a word, otherwise we won't use it.

5. "at least programmers are willing to occasionally refresh our idioms and notations to better reflect our mutual understanding." Mathematicians do this all the time, not just "occasionally". It doesn't happen in mature and well established topics where all the problems are dry. There are nonstandard analysis, complementary to the standard analysis. There are homotopy type theory to have another foundation for mathematics. Many things we seem to see it and think it's set into stone it's because there really isn't anything to be done.